An extremal property of Fekete polynomials

Abstract

The Fekete polynomials are defined as \[ F_q(z) := \sum ^{q-1}_{k=1} \left (\frac {k}{q}\right ) z^k\] where $\left (\frac {\cdot }{q}\right )$ is the Legendre symbol. These polynomials arise in a number of contexts in analysis and number theory. For example, after cyclic permutation they provide sequences with smallest known $L_4$ norm out of the polynomials with $\pm 1$ coefficients. The main purpose of this paper is to prove the following extremal property that characterizes the Fekete polynomials by their size at roots of unity. Theorem 0.1. Let $f(x)=a_1x+a_2x^2+\cdots +a_{N-1}x^{N-1}$ with odd $N$ and $a_n=\pm 1$. If \[ \operatorname {max}\{ |f(\omega ^k)| : 0 \le k \le N-1 \} = \sqrt {N}, \] then $N$ must be an odd prime and $f(x)$ is $\pm F_q(x)$. Here $\omega :=e^{\frac {2\pi i}{N}}.$ This result also gives a partial answer to a problem of Harvey Cohn on character sums.